Any ideas on places that my mother (75) can live?

I don’t know what state you’re in, but you might inquire from your mother’s department of social services. In my state, NC, one may apply to their department of social services for all help, county, state and federal. Some states have assistance for those who qualify financially AND need assistance, as documented by a doctor.It’s called Special Assistance and is similar to Medicaid. People with dementia who qualify, may get help to live in AL or MC. It’s my understanding that not all states have this benefit. If she already has Medicaid, her assigned case manager should be able to direct you. I’d try to figure a way to get her evaluated.

I agree with Geaton that it's important now for the future that someone can act for her when she is unable. You say she "likely" has dementia. Would she sign papers to make you her POA? She would likely still pass the Lawyer's tests if she wishes to do this, as to what she is doing and why and who she wishes to handle things for her if needed. I would approach this as "We should see a lawyer so that if you are ever in need of help I can get that help for you; I can act for you if you are unable, and etc".
If she can't do this, eventually you will end in a situation where there is a hospitalization and a social worker who will assist you with finding placement. They are much better at that than we are, especially with medicaid being the payment.
If worst comes to worst Mom will end taken to the ER for evaluation. That is the time to call in the heavy duty help. I think you will have a hard time finding placement with her income, as facilities do try to "cherry pick"; they are businesses.

Kate, are you your mom's durable PoA? If not you won't be legally able to make her move unless you pursue guardianship through the courts. If you do have the legal documents as her PoA then I would get her tested. You can employ a "therapeutic fib" to get her to the doctor, such as: Medicare now requires you to have an annual physical". When there ask for their Medical Representative form and have her write your name and she signs it. This allows the docs at that office to discuss her info with you without her being present or requiring her permission (HIPAA form). Then pass the staff a pre-written note asking them to perform a cognitive exam and test for a UTI (they will be happy to do this). If she is diagnosed with cognitive impairment then your PoA may be able to be activated (it depends on what the PoA docs specify). You are correct in thinking public housing won't be safe for her... please contact social services at her county's Dept of Health and Human Services to ask about assistance for AL. Also contact her local area's Agency on Aging for other resources and guidance.